We begin by briefly summarizing the contents of this paper; details, and some definitions of terminology, appear in subsequent sections. All hypothesized topological spaces are assumed to be Hausdorff. The reader is referred to [13] for undefined notation and terminology.

A perfect irreducible continuous surjection is called a covering map. Let Xbe a space, let f and g be two such functions with domain X,and let Rf denote the range of (i.e., the set f [Z]). Then f and g are said to be equivalent (denoted f≈ g) if there is a homeomorphism h : Rf —” Rg such that h of = g. We identify equivalent covering maps with domain X, and then denote by IP(X)the set of such covering maps.

For each Hausdorff space X, let FX be an Hausdorff extension of X. The existence of the largest subcategory of HAUS on which F s a functor and an epi-reflection is investigated.

Let X be a metric space. Assume either that X is locally compact or that X has no more than countably many isolated points. It is proved that if F is a nowhere dense subset of X, then it is regularly nowhere dense (in the sense of Katětov) and hence is contained in the topological boundary of some regular-closed subset of X. This result is used to obtain new properties of the remote points of the Stone-Čech compactification of a metric space without isolated points.

Let βX denote the Stone-Čech compactification of the completely regular Hausdorff space X. Fine and Gillman [3] define a point p of βX to be remote if p is not in the βX-closure of a discrete subset of X.

A topological space is E0 (resp. E1) provided every point is the countable intersection of neighborhoods (resp. closed neighborhoods). For i = 0 and i = 1, characterizations of minimal Ei. spaces (Ei. spaces with no strictly coarser Ei. topology) and Ei-closed spaces (Ei. spaces which are closed in every Ei. space containing them) are given; for example, the properties of minimal Ei. and minimal first countable Ti+1 are shown to be equivalent. Minimal E0 spaces are characterized as countable spaces with the cofinite topology, and minimal E1 spaces are characterized as E1-closed and semiregular spaces. E0-closed spaces are shown to be precisely the finite discrete spaces.